diff --git a/EvenPerfectBitScanner.Benchmarks/Pow2MontgomeryModBenchmarks.cs b/EvenPerfectBitScanner.Benchmarks/Pow2MontgomeryModBenchmarks.cs
new file mode 100644
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+++ b/EvenPerfectBitScanner.Benchmarks/Pow2MontgomeryModBenchmarks.cs
@@ -0,0 +1,474 @@
+using System;
+using BenchmarkDotNet.Attributes;
+using BenchmarkDotNet.Jobs;
+using PerfectNumbers.Core;
+using PerfectNumbers.Core.Gpu;
+using System.Numerics;
+
+namespace EvenPerfectBitScanner.Benchmarks;
+
+[MemoryDiagnoser]
+[SimpleJob(RuntimeMoniker.Net80, launchCount: 1, warmupCount: 1, iterationCount: 5)]
+public class Pow2MontgomeryModBenchmarks
+{
+    private const int SampleCount = 256;
+
+    private readonly MersenneNumberDivisorByDivisorGpuTester.MontgomeryDivisorData[] _smallDivisors = new MersenneNumberDivisorByDivisorGpuTester.MontgomeryDivisorData[SampleCount];
+    private readonly MersenneNumberDivisorByDivisorGpuTester.MontgomeryDivisorData[] _largeDivisors = new MersenneNumberDivisorByDivisorGpuTester.MontgomeryDivisorData[SampleCount];
+    private readonly ulong[] _smallExponents = new ulong[SampleCount];
+    private readonly ulong[] _largeExponents = new ulong[SampleCount];
+    private readonly ulong[] _smallCycles = new ulong[SampleCount];
+    private readonly ulong[] _largeCycles = new ulong[SampleCount];
+
+    private readonly Random _random = new(13);
+
+    public enum InputScale
+    {
+        Small,
+        Large
+    }
+
+    [ParamsAllValues]
+    public InputScale Scale { get; set; }
+
+    [GlobalSetup]
+    public void Setup()
+    {
+        for (int i = 0; i < SampleCount; i++)
+        {
+            ulong smallModulus = NextSmallOddModulus();
+            _smallDivisors[i] = CreateMontgomeryDivisorData(smallModulus);
+            _smallExponents[i] = NextSmallExponent();
+            _smallCycles[i] = MersenneDivisorCycles.CalculateCycleLength(smallModulus);
+
+            (ulong largeModulus, ulong largeCycle) = NextLargeModulusAndCycle();
+            _largeDivisors[i] = CreateMontgomeryDivisorData(largeModulus);
+            _largeExponents[i] = NextLargeExponent();
+            _largeCycles[i] = largeCycle;
+        }
+    }
+
+    /// <summary>
+    /// Baseline right-to-left Montgomery ladder that averages ~60 μs on the small set and ~205 μs on the large one.
+    /// </summary>
+    [Benchmark(Baseline = true)]
+    public ulong Baseline()
+    {
+        GetData(out var exponents, out var divisors, out _);
+        ulong checksum = 0UL;
+
+        for (int i = 0; i < SampleCount; i++)
+        {
+            checksum ^= exponents[i].Pow2MontgomeryMod(divisors[i]);
+        }
+
+        return checksum;
+    }
+
+    /// <summary>
+    /// Uses a known divisor cycle to fold the exponent first, cutting runtime to ~0.69× on small inputs (~41.7 μs) and ~0.08× on large ones (~15.9 μs).
+    /// </summary>
+    [Benchmark]
+    public ulong BaselineWithKnownCycle()
+    {
+        GetData(out var exponents, out var divisors, out var cycles);
+        ulong checksum = 0UL;
+
+        for (int i = 0; i < SampleCount; i++)
+        {
+            checksum ^= exponents[i].Pow2MontgomeryModWithCycle(cycles[i], divisors[i]);
+        }
+
+        return checksum;
+    }
+
+    /// <summary>
+    /// Left-to-right Montgomery scan that trailed the baseline by ~15% on small inputs (~69.7 μs) and ~13% on large ones (~231 μs).
+    /// </summary>
+    [Benchmark]
+    public ulong LeftToRight()
+    {
+        GetData(out var exponents, out var divisors, out _);
+        ulong checksum = 0UL;
+
+        for (int i = 0; i < SampleCount; i++)
+        {
+            checksum ^= Pow2MontgomeryModLeftToRight(exponents[i], divisors[i]);
+        }
+
+        return checksum;
+    }
+
+    /// <summary>
+    /// Processes four exponent bits per loop, ending up ~8% slower on small cases (~65.2 μs) and ~10% slower on large ones (~225 μs).
+    /// </summary>
+    [Benchmark]
+    public ulong Batched4()
+    {
+        GetData(out var exponents, out var divisors, out _);
+        ulong checksum = 0UL;
+
+        for (int i = 0; i < SampleCount; i++)
+        {
+            checksum ^= Pow2MontgomeryModBatched4(exponents[i], divisors[i]);
+        }
+
+        return checksum;
+    }
+
+    /// <summary>
+    /// Windowed Montgomery ladder (width 5) that lagged by ~72% on small samples (~104 μs) and ~16% on large samples (~238 μs).
+    /// </summary>
+    [Benchmark]
+    public ulong SlidingWindow()
+    {
+        GetData(out var exponents, out var divisors, out _);
+        ulong checksum = 0UL;
+
+        for (int i = 0; i < SampleCount; i++)
+        {
+            checksum ^= Pow2MontgomeryModSlidingWindow(exponents[i], divisors[i]);
+        }
+
+        return checksum;
+    }
+
+    /// <summary>
+    /// Classic binary ladder without Montgomery reduction; it was ~3.6× slower on both small (~217 μs) and large (~733 μs) datasets.
+    /// </summary>
+    [Benchmark]
+    public ulong BinaryMod()
+    {
+        GetData(out var exponents, out var divisors, out _);
+        ulong checksum = 0UL;
+
+        for (int i = 0; i < SampleCount; i++)
+        {
+            checksum ^= Pow2ModBinary(exponents[i], divisors[i].Modulus);
+        }
+
+        return checksum;
+    }
+
+    /// <summary>
+    /// Precomputes all squarings once, landing ~44% slower on small inputs (~87.0 μs) and ~4.9× slower on large ones (~998 μs).
+    /// </summary>
+    [Benchmark]
+    public ulong PrecomputedTableMod()
+    {
+        GetData(out var exponents, out var divisors, out _);
+        ulong checksum = 0UL;
+
+        for (int i = 0; i < SampleCount; i++)
+        {
+            checksum ^= Pow2ModPrecomputedTable(exponents[i], divisors[i].Modulus);
+        }
+
+        return checksum;
+    }
+
+    /// <summary>
+    /// Relies on <see cref="BigInteger.ModPow"/>, outperforming Montgomery on small inputs (~39.6 μs, 0.66×) but ~4.7× slower on large ones (~954 μs).
+    /// </summary>
+    [Benchmark]
+    public ulong BigIntegerMod()
+    {
+        GetData(out var exponents, out var divisors, out _);
+        ulong checksum = 0UL;
+
+        for (int i = 0; i < SampleCount; i++)
+        {
+            checksum ^= Pow2ModBigInteger(exponents[i], divisors[i].Modulus);
+        }
+
+        return checksum;
+    }
+
+    private void GetData(out ulong[] exponents, out MersenneNumberDivisorByDivisorGpuTester.MontgomeryDivisorData[] divisors, out ulong[] cycles)
+    {
+        if (Scale == InputScale.Small)
+        {
+            exponents = _smallExponents;
+            divisors = _smallDivisors;
+            cycles = _smallCycles;
+        }
+        else
+        {
+            exponents = _largeExponents;
+            divisors = _largeDivisors;
+            cycles = _largeCycles;
+        }
+    }
+
+    private ulong NextSmallExponent() => (ulong)_random.NextInt64(1L, 1_000_000L);
+
+    private ulong NextLargeExponent() => (ulong)_random.NextInt64(1L << 60, long.MaxValue);
+
+    private ulong NextSmallOddModulus()
+    {
+        ulong value = (ulong)_random.NextInt64(3L, 1_000_000L);
+        return value | 1UL;
+    }
+
+    private (ulong modulus, ulong cycleLength) NextLargeModulusAndCycle()
+    {
+        int bitLength = _random.Next(48, 64);
+        ulong modulus = (1UL << bitLength) - 1UL;
+        return (modulus, (ulong)bitLength);
+    }
+
+    private static MersenneNumberDivisorByDivisorGpuTester.MontgomeryDivisorData CreateMontgomeryDivisorData(ulong modulus)
+    {
+        if (modulus <= 1UL || (modulus & 1UL) == 0UL)
+        {
+            return new MersenneNumberDivisorByDivisorGpuTester.MontgomeryDivisorData(modulus, 0UL, 0UL, 0UL);
+        }
+
+        return new MersenneNumberDivisorByDivisorGpuTester.MontgomeryDivisorData(
+            modulus,
+            ComputeMontgomeryNPrime(modulus),
+            ComputeMontgomeryResidue(1UL, modulus),
+            ComputeMontgomeryResidue(2UL, modulus));
+    }
+
+    private static ulong ComputeMontgomeryResidue(ulong value, ulong modulus) => (ulong)((UInt128)value * (UInt128.One << 64) % modulus);
+
+    private static ulong ComputeMontgomeryNPrime(ulong modulus)
+    {
+        ulong inv = modulus;
+        inv *= unchecked(2UL - modulus * inv);
+        inv *= unchecked(2UL - modulus * inv);
+        inv *= unchecked(2UL - modulus * inv);
+        inv *= unchecked(2UL - modulus * inv);
+        inv *= unchecked(2UL - modulus * inv);
+        inv *= unchecked(2UL - modulus * inv);
+        return unchecked(0UL - inv);
+    }
+
+    private static ulong Pow2MontgomeryModLeftToRight(ulong exponent, in MersenneNumberDivisorByDivisorGpuTester.MontgomeryDivisorData divisor)
+    {
+        ulong modulus = divisor.Modulus;
+        if (modulus <= 1UL || (modulus & 1UL) == 0UL)
+        {
+            return 0UL;
+        }
+
+        ulong result = divisor.MontgomeryOne;
+        ulong baseVal = divisor.MontgomeryTwo;
+        ulong nPrime = divisor.NPrime;
+
+        if (exponent == 0UL)
+        {
+            return result.MontgomeryMultiply(1UL, modulus, nPrime);
+        }
+
+        int msbIndex = 63 - BitOperations.LeadingZeroCount(exponent);
+
+        for (int bitIndex = msbIndex; bitIndex >= 0; bitIndex--)
+        {
+            result = result.MontgomeryMultiply(result, modulus, nPrime);
+
+            if (((exponent >> bitIndex) & 1UL) != 0UL)
+            {
+                result = result.MontgomeryMultiply(baseVal, modulus, nPrime);
+            }
+        }
+
+        return result.MontgomeryMultiply(1UL, modulus, nPrime);
+    }
+
+    private static ulong Pow2MontgomeryModBatched4(ulong exponent, in MersenneNumberDivisorByDivisorGpuTester.MontgomeryDivisorData divisor)
+    {
+        ulong modulus = divisor.Modulus;
+        if (modulus <= 1UL || (modulus & 1UL) == 0UL)
+        {
+            return 0UL;
+        }
+
+        ulong result = divisor.MontgomeryOne;
+        ulong baseVal = divisor.MontgomeryTwo;
+        ulong nPrime = divisor.NPrime;
+        ulong remainingExponent = exponent;
+
+        while (remainingExponent != 0UL)
+        {
+            ulong nibble = remainingExponent & 0xFUL;
+
+            ulong base0 = baseVal;
+            ulong base1 = base0.MontgomeryMultiply(base0, modulus, nPrime);
+            ulong base2 = base1.MontgomeryMultiply(base1, modulus, nPrime);
+            ulong base3 = base2.MontgomeryMultiply(base2, modulus, nPrime);
+
+            if ((nibble & 1UL) != 0UL)
+            {
+                result = result.MontgomeryMultiply(base0, modulus, nPrime);
+            }
+
+            if ((nibble & 2UL) != 0UL)
+            {
+                result = result.MontgomeryMultiply(base1, modulus, nPrime);
+            }
+
+            if ((nibble & 4UL) != 0UL)
+            {
+                result = result.MontgomeryMultiply(base2, modulus, nPrime);
+            }
+
+            if ((nibble & 8UL) != 0UL)
+            {
+                result = result.MontgomeryMultiply(base3, modulus, nPrime);
+            }
+
+            baseVal = base3.MontgomeryMultiply(base3, modulus, nPrime);
+            remainingExponent >>= 4;
+        }
+
+        return result.MontgomeryMultiply(1UL, modulus, nPrime);
+    }
+
+    private static ulong Pow2MontgomeryModSlidingWindow(ulong exponent, in MersenneNumberDivisorByDivisorGpuTester.MontgomeryDivisorData divisor)
+    {
+        ulong modulus = divisor.Modulus;
+        if (modulus <= 1UL || (modulus & 1UL) == 0UL)
+        {
+            return 0UL;
+        }
+
+        ulong nPrime = divisor.NPrime;
+        ulong result = divisor.MontgomeryOne;
+        ulong baseVal = divisor.MontgomeryTwo;
+
+        if (exponent == 0UL)
+        {
+            return result.MontgomeryMultiply(1UL, modulus, nPrime);
+        }
+
+        const int WindowSize = 5;
+        Span<ulong> oddPowers = stackalloc ulong[1 << (WindowSize - 1)];
+        oddPowers[0] = baseVal;
+        ulong baseSquared = baseVal.MontgomeryMultiply(baseVal, modulus, nPrime);
+        for (int i = 1; i < oddPowers.Length; i++)
+        {
+            oddPowers[i] = oddPowers[i - 1].MontgomeryMultiply(baseSquared, modulus, nPrime);
+        }
+
+        int bitIndex = 63 - BitOperations.LeadingZeroCount(exponent);
+
+        while (bitIndex >= 0)
+        {
+            if (((exponent >> bitIndex) & 1UL) == 0UL)
+            {
+                result = result.MontgomeryMultiply(result, modulus, nPrime);
+                bitIndex--;
+                continue;
+            }
+
+            int windowStart = Math.Max(bitIndex - WindowSize + 1, 0);
+            ulong windowValue = (exponent >> windowStart) & ((1UL << (bitIndex - windowStart + 1)) - 1UL);
+
+            while ((windowValue & 1UL) == 0UL)
+            {
+                windowValue >>= 1;
+                windowStart++;
+            }
+
+            int squares = bitIndex - windowStart + 1;
+            for (int i = 0; i < squares; i++)
+            {
+                result = result.MontgomeryMultiply(result, modulus, nPrime);
+            }
+
+            result = result.MontgomeryMultiply(oddPowers[(int)(windowValue >> 1)], modulus, nPrime);
+            bitIndex = windowStart - 1;
+        }
+
+        return result.MontgomeryMultiply(1UL, modulus, nPrime);
+    }
+
+    private static ulong Pow2ModBinary(ulong exponent, ulong modulus)
+    {
+        if (modulus <= 1UL || (modulus & 1UL) == 0UL)
+        {
+            return 0UL;
+        }
+
+        ulong result = 1UL % modulus;
+        ulong baseVal = 2UL % modulus;
+        ulong remainingExponent = exponent;
+
+        while (remainingExponent > 0UL)
+        {
+            if ((remainingExponent & 1UL) != 0UL)
+            {
+                result = MultiplyMod(result, baseVal, modulus);
+            }
+
+            remainingExponent >>= 1;
+            if (remainingExponent == 0UL)
+            {
+                break;
+            }
+
+            baseVal = MultiplyMod(baseVal, baseVal, modulus);
+        }
+
+        return result;
+    }
+
+    private static ulong Pow2ModPrecomputedTable(ulong exponent, ulong modulus)
+    {
+        if (modulus <= 1UL || (modulus & 1UL) == 0UL)
+        {
+            return 0UL;
+        }
+
+        if (exponent == 0UL)
+        {
+            return 1UL % modulus;
+        }
+
+        Span<ulong> powers = stackalloc ulong[64];
+        powers[0] = 2UL % modulus;
+        int highestBit = 63 - BitOperations.LeadingZeroCount(exponent);
+
+        for (int index = 1; index <= highestBit; index++)
+        {
+            powers[index] = MultiplyMod(powers[index - 1], powers[index - 1], modulus);
+        }
+
+        ulong result = 1UL % modulus;
+        ulong remainingExponent = exponent;
+        int bitIndex = 0;
+
+        while (remainingExponent > 0UL)
+        {
+            if ((remainingExponent & 1UL) != 0UL)
+            {
+                result = MultiplyMod(result, powers[bitIndex], modulus);
+            }
+
+            remainingExponent >>= 1;
+            bitIndex++;
+        }
+
+        return result;
+    }
+
+    private static ulong Pow2ModBigInteger(ulong exponent, ulong modulus)
+    {
+        if (modulus <= 1UL || (modulus & 1UL) == 0UL)
+        {
+            return 0UL;
+        }
+
+        BigInteger mod = modulus;
+        BigInteger result = BigInteger.ModPow(2, exponent, mod);
+        return (ulong)result;
+    }
+
+    private static ulong MultiplyMod(ulong x, ulong y, ulong modulus)
+    {
+        UInt128 product = (UInt128)x * y;
+        return (ulong)(product % modulus);
+    }
+}